#230437

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,6,3,7,5,4,8)(9,10)(11,21,25,35)(12,13)(14,19)(15,26)(16,20)(17,24,36,32)(18,27)(22,37,39,23)(28,31)(29,38,40,34)(30,33);(1,3)(2,6)(4,5)(7,8)(9,11,15,24,16,25,14,32)(10,17,26,21,20,36,19,35)(12,22,33,38,28,39,27,34)(13,29,30,37,31,40,18,23);(1,4,7,6)(2,3,5,8)(9,12,16,28)(10,18,20,30)(11,22,25,39)(13,19,31,26)(14,27,15,33)(17,23,36,37)(21,29,35,40)(24,38,32,34);(1,5,4,3,7,2,6,8)(9,13,15,30,16,31,14,18)(10,12,26,33,20,28,19,27)(11,23,32,40,25,37,24,29)(17,34,35,39,36,38,21,22)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137633);